Fill a large stockpot with 3 quarts of water. Squeeze the juice from the lemon halves into the water, then add the lemon halves, Old Bay, kosher salt, and bay leaf. Bring to a rolling boil.
Add the potatoes and boil for 12 to 15 minutes (depending on their size) until they are nearly fork-tender.
Add the corn and cook for 4 to 5 minutes.
Add the shrimp and cook for 2 to 3 minutes, or until they are pink, opaque, and just cooked through.
Carefully drain the boil and discard the lemon halves and bay leaf.
Return the potatoes, corn, and shrimp to the pot. Pour about two-thirds of the Cajun butter sauce over everything and gently toss until well coated.
Transfer the shrimp boil to a large serving platter or spread it onto a parchment-lined table. Drizzle with the remaining Cajun butter sauce and sprinkle with fresh parsley. Serve with lemon wedges.